We study periods of multi-parameter families of K3 surfaces, which are relevant to compute the maximal cuts of certain classes of Feynman integrals. We focus on their automorphic properties, and we show that generically the periods define orthogonal modular forms. Using accidental isomorphisms between Lie groups of small rank, we show how one can use the intersection product on the periods to identify K3 surfaces whose periods can be expressed in terms of other classes of modular forms that have been studied in the mathematics literature. We apply our results to maximal cuts of three-loop banana integrals, and we show that depending on the mass configuration, the maximal cuts define ordinary modular forms or Hilbert, Siegel or hermitian modular forms.

Germany

Claude Duhr

Modular forms for three-loop banana integrals

We compute the contributions of CP-conserving operators in the dimension-six SMEFT to the two-loop renormalisation constants of quark and gluon fields in the $$ \overline{\textrm{MS}} $$ -scheme. Specifically, we consider the top-quark chromomagnetic operator and the triple gluon operator. We work with the background-field method, which allows us to extract the contribution of these operators to the two-loop running of the top mass and the strong coupling constant. We discuss in detail the mixing with the unphysical operators required for the renormalisation, and we present analytic results for the renormalisation constants of all relevant operators.

Andres Vasquez

United Kingdom

Giuseppe Ventura

Eleni Vryonidou

Two-loop renormalisation of quark and gluon fields in the SMEFT

The three-loop banana integral with three equal masses and the conformal two-loop five-point traintrack integral in two dimensions are related to a two-parameter family of K3 surfaces. We compute the corresponding periods and the mirror map, and we show that they can be expressed in terms of ordinary modular forms and functions. In particular, we find that the maximal cuts of the three-loop banana integral with three equal masses can be written as a product of two copies of the maximal cuts of the two-loop equal-mass sunrise integral. Our computation reveals a hidden symmetry of the banana integral not manifest from the Feynman integral representation, which corresponds to exchanging the two copies of the sunrise elliptic curve.

Sara Maggio

Feynman integrals, elliptic integrals and two-parameter K3 surfaces

We show how a method to construct canonical differential equations for multi-loop Feynman integrals recently introduced by some of the authors can be extended to cases where the associated geometry is of Calabi-Yau type and even beyond. This can be achieved by supplementing the method with information from the mixed Hodge structure of the underlying geometry. We apply these ideas to specific classes of integrals whose associated geometry is a one-parameter family of Calabi-Yau varieties, and we argue that the method can always be successfully applied to those cases. Moreover, we perform an in-depth study of the properties of the resulting canonical differential equations. In particular, we show that the resulting canonical basis is equivalent to the one obtained by an alternative method recently introduced in the literature. We apply our method to non-trivial and cutting-edge examples of Feynman integrals necessary for gravitational wave scattering, further showcasing its power and flexibility.

Christoph Nega

Benjamin Sauer

Lorenzo Tancredi

Fabian Wagner

Aspects of canonical differential equations for Calabi-Yau geometries and beyond

<p>In strongly deformed planar <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> super-Yang-Mills theory, or fishnet theory, a point-split single-trace correlation function of four dimension-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi></math> scalar operators is given by a single Feynman integral, which involves integrating over locations of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi><mo>×</mo><mi>m</mi></math> grid of points. We show that for any integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>m</mi></math> this square fishnet graph is invariant under the combined action of a kinematic map and the antipode map of the Hopf algebra on multiple polylogarithms; i.e. it possesses an antipodal self-duality.</p>

United States

Lance J. Dixon

Antipodal self-duality of square fishnet graphs

Recently a notion of self-duality for differential equations of maximal cuts was introduced, which states that there should be a basis in which the matrix for an -factorised differential equation is persymmetric. It was observed that the rotation to this special basis may introduce a Galois symmetry relating different integrals. We argue that the proposed notion of self-duality for maximal cuts stems from a very natural notion of self-duality from twisted cohomology. Our main result is that, if the differential equations and their duals are simultaneously brought into canonical form, the cohomology intersection matrix is a constant. Furthermore, we show that one can associate quite generically a Lie algebra representation to an -factorised system. For maximal cuts, this representation is irreducible and self-dual. The constant intersection matrix can be interpreted as expressing the equivalence of this representation and its dual, which in turn results in constraints for the differential equation matrix. Unlike the earlier proposal, the most natural symmetry of the differential equation matrix is defined entirely over the rational numbers and is independent of the basis choice.

Franziska Porkert

Cathrin Semper

Sven Stawinski

Self-duality from twisted cohomology

Using the framework of twisted cohomology, we study twisted Riemann bilinear relations (TRBRs) satisfied by multi-loop Feynman integrals and their cuts in dimensional regularisation. After showing how to associate to a given family of Feynman integrals a period matrix whose entries are cuts, we investigate the TRBRs satisfied by this period matrix, its dual and the intersection matrices for twisted cycles and co-cycles. For maximal cuts, the non-relative framework is applicable, and the period matrix and its dual are related in a simple manner. We then find that the TRBRs give rise to quadratic relations that generalise quadratic relations that have previously appeared in the literature. However, we find that the TRBRs do not allow us to obtain quadratic relations for non-maximal cuts or completely uncut Feynman integrals. This can be traced back to the fact that the TRBRs are not quadratic in the period matrix, but separately linear in the period matrix and its dual, and the two are not simply related in the case of a relative cohomology theory, which is required for non-maximal cuts.

Twisted Riemann bilinear relations and Feynman integrals

We discuss for the first time canonical differential equations for hyperelliptic Feynman integrals. We study hyperelliptic Lauricella functions that include in particular the maximal cut of the two-loop non-planar double box, which is known to involve a hyperlliptic curve of genus two. We consider specifically three- and four-parameter Lauricella functions, each associated to a hyperelliptic curve of genus two, and construct their canonical differential equations. Whilst core steps of this construction rely on existing methods — that we show to be applicable in the higher-genus case — we use new ideas on the structure of the twisted cohomology intersection matrix associated to the integral family in canonical form to obtain a better understanding of the appearing new functions. We further observe the appearance of Siegel modular forms in the -factorized differential equation matrix, nicely generalizing similar observations from the elliptic case.

Canonical differential equations beyond genus one

We compute the electron self-energy in Quantum Electrodynamics to three loops in terms of iterated integrals over kernels of elliptic type. We make use of the differential equations method, augmented by an $\textit{ϵ}$-factorized basis, which allows us to gain full control over the differential forms appearing in the iterated integrals to all orders in the dimensional regulator. We obtain compact analytic expressions, for which we provide generalized series expansion representations that allow us to evaluate the result numerically for all values of the electron momentum squared. As a by-product, we also obtain $\textit{ϵ}$-resummed results for the self-energy in the on-shell limit $\textit{p}$ = $\textit{m}$ , which we use to recompute the known three-loop renormalization constants in the on-shell scheme.

Federico Gasparotto

Stefan Weinzierl

On the electron self-energy to three loops in QED

Numerical approaches to computations typically reconstruct the numerators of Feynman diagrams in four dimensions. In doing so, certain rational terms arising from the (D − 4)-dimensional part of the numerator multiplying ultraviolet (UV) poles in dimensional regularisation are not captured and need to be obtained by other means. At one-loop these rational terms of UV origin can be computed from a set of process-independent Feynman rules. Recently, it was shown that this approach can be extended to two loops. In this paper, we show that to all loop orders it is possible to compute rational terms of UV origin through process-independent vertices that are polynomial in masses and momenta.